Let $X$ be a scheme of finite type over an algebraically closed field $k$. Then the topological space of $X$ is comprised of all $k$-points, i.e all morphism $\operatorname{Spec} k \to X$.
Now suppose that $X$ is a scheme over $S$. I am trying to recover the topological space $|X|$ of $X$ from the functor of points interpretation of $X$, that is what schemes fits into the following equality $|X|= \operatorname{Hom}_{Sch}(?, X)$?
If you're just interested in the underlying set $X$ then you can consider the following generalization of the $k$-points construction. Let us say that two maps $\mathrm{Spec}(K_1)\to X$ and $\mathrm{Spec}(K_2)\to X$ where $K_i$ are fields are called equivalent if there exists a third field $L$ and maps $K_i\to L$ such that the compositions $\mathrm{Spec}(L)\to\mathrm{Spec}(K_i)\to X$ agree. Then, you can form the set $\{\mathrm{Spec}(K)\to X\}/\sim$ where $K$ ranges over all fields. This set can be identified with the underlying set of $X$ as a topological space.